Civil Engineering Reference
In-Depth Information
However, this is in contradiction to the definition of the transmission coefficient
since the incident power is independent of edge effects, contrary to the input power.
It is recommended here that the correction be only applied to the transmitted power.
Moreover, this correlates well with TL measurement. However, a correction may still be
necessary to account for the diffusiveness of the incident field.
12.3
Finite size correction for the absorption problem
Similarly to the transmission loss problem, a simple way to correct the statistical absorp-
tion coefficient (obtained for a material of infinite extent) is presented in order to get
a better match between numerical and reverberant room experimental results. The pre-
sented formulation is similar to the one presented in Thomasson (1980) and uses the
geometrical radiation impedance introduced in the previous section.
12.3.1 Surface pressure
For a plane incident wave with heading angles
(θ,φ)
, the acoustic pressure in the emitter
(source) domain reads
p(M)
=
p
b
(M)
+
p
ray
(M)
(12.20)
where
p
b
(M)
is the blocked pressure given by
p
b
(M)
=
exp[
−
jk
0
(
cos
φ
sin
θx
+
sin
φ
sin
θy
+
cos
θz)
]
−
cos
θz)
] (12.21)
On the surface
S
(
z
=
0),
∂ p
b
(x,y,
0
)
∂z
=
0and
p
b
(x,y,
0
)
=
2
p
i
(x,y,
0
)
.
Assuming that the material has a space-independent normalized admittance
β
=
+
exp[
−
jk
0
(
cos
φ
sin
θx
+
sin
φ
sin
θy
ρ
0
c
Z
m
where
Z
m
is the surface impedance of the material, the impedance condition at
the surface reads
∂ p
∂n
in
jkβ p
where
n
in
is the surface normal vector pointing into
the material. Rewriting this relation with the outward normal pointing into the emitter
medium, the radiated pressure given by Equation (12.3) becomes
=−
jkβ
p
ray
(M)
=−
p(M
0
)G(M,M
0
)
d
S(M
0
)
(12.22)
S
which reads, on the surface of the material
jkβ
p(M)
=
2
p
i
(M)
−
p(M
0
)G(M,M
0
)
d
S(M
0
)
(12.23)
S
B p
i
=
(
1
+
V
f
) p
i
where
B
is assumed constant over the surface.
V
f
is the equivalent reflection coefficient
accounting for the finite size of the sample. It is referred to here by the subscript
f
in
contrast with the classical 'infinite size' reflection coefficient
V
∞
.
Consider
δ p
=
p
i
δ B
an admissible variation of the parietal pressure field. Multiplying
Equation (12.23) by the complex conjugate of this variation and integrating over the
The pressure at the surface can be written under the form:
p
=
